Distorted Schwarzschild metric

All static axisymmetric solutions of the Einstein–Maxwell equations can be written in the form of Weyl's metric,[1]

${\displaystyle (1)\quad ds^{2}=-e^{2\psi (\rho ,z)}dt^{2}+e^{2\gamma (\rho ,z)-2\psi (\rho ,z)}(d\rho ^{2}+dz^{2})+e^{-2\psi (\rho ,z)}\rho ^{2}d\phi ^{2}\,,}$

From the Weyl perspective, the metric potentials generating the standard Schwarzschild solution are given by[1][2]

${\displaystyle (2)\quad \psi _{SS}={\frac {1}{2}}\ln {\frac {L-M}{L+M}}\,,\quad \gamma _{SS}={\frac {1}{2}}\ln {\frac {L^{2}-M^{2}}{l_{+}l_{-}}}\,,}$

where

${\displaystyle (3)\quad L={\frac {1}{2}}{\big (}l_{+}+l_{-}{\big )}\,,\quad l_{+}={\sqrt {\rho ^{2}+(z+M)^{2}}}\,,\quad l_{-}={\sqrt {\rho ^{2}+(z-M)^{2}}}\,,}$

which yields the Schwarzschild metric in Weyl's canonical coordinates that

${\displaystyle (4)\quad ds^{2}=-{\frac {L-M}{L+M}}dt^{2}+{\frac {(L+M)^{2}}{l_{+}l_{-}}}(d\rho ^{2}+dz^{2})+{\frac {L+M}{L-M}}\,\rho ^{2}d\phi ^{2}\,.}$

Vacuum Weyl spacetimes (such as Schwarzschild) respect the following field equations,[1][2]

${\displaystyle (5.a)\quad \nabla ^{2}\psi =0\,,}$

${\displaystyle (5.b)\quad \gamma _{,\,\rho }=\rho \,{\Big (}\psi _{,\,\rho }^{2}-\psi _{,\,z}^{2}{\Big )}\,,}$

${\displaystyle (5.c)\quad \gamma _{,\,z}=2\,\rho \,\psi _{,\,\rho }\psi _{,\,z}\,,}$

${\displaystyle (5.d)\quad \gamma _{,\,\rho \rho }+\gamma _{,\,zz}=-{\big (}\psi _{,\,\rho }^{2}+\psi _{,\,z}^{2}{\big )}\,,}$

where ${\displaystyle \nabla ^{2}:=\partial _{\rho \rho }+{\frac {1}{\rho }}\partial _{\rho }+\partial _{zz}}$ is the Laplace operator.

Eq(5.a) is the linear Laplace's equation; that is to say, linear combinations of given solutions are still its solutions. Given two solutions ${\displaystyle \{\psi ^{\langle 1\rangle },\psi ^{\langle 2\rangle }\}}$ to Eq(5.a), one can construct a new solution via

${\displaystyle (6)\quad {\tilde {\psi }}\,=\,\psi ^{\langle 1\rangle }+\psi ^{\langle 2\rangle }\,,}$

and the other metric potential can be obtained by

${\displaystyle (7)\quad {\tilde {\gamma }}\,=\,\gamma ^{\langle 1\rangle }+\gamma ^{\langle 2\rangle }+2\int \rho \,{\Big \{}\,{\Big (}\psi _{,\,\rho }^{\langle 1\rangle }\psi _{,\,\rho }^{\langle 2\rangle }-\psi _{,\,z}^{\langle 1\rangle }\psi _{,\,z}^{\langle 2\rangle }{\Big )}\,d\rho +{\Big (}\psi _{,\,\rho }^{\langle 1\rangle }\psi _{,\,z}^{\langle 2\rangle }+\psi _{,\,z}^{\langle 1\rangle }\psi _{,\,\rho }^{\langle 2\rangle }{\Big )}\,dz\,{\Big \}}\,.}$

Let ${\displaystyle \psi ^{\langle 1\rangle }=\psi _{SS}}$ and ${\displaystyle \gamma ^{\langle 1\rangle }=\gamma _{SS}}$, while ${\displaystyle \psi ^{\langle 2\rangle }=\psi }$ and ${\displaystyle \gamma ^{\langle 2\rangle }=\gamma }$ refer to a second set of Weyl metric potentials. Then, ${\displaystyle \{{\tilde {\psi }},{\tilde {\gamma }}\}}$ constructed via Eqs(6)(7) leads to the superposed Schwarzschild-Weyl metric

${\displaystyle (8)\quad ds^{2}=-e^{2\psi (\rho ,z)}{\frac {L-M}{L+M}}dt^{2}+e^{2\gamma (\rho ,z)-2\psi (\rho ,z)}{\frac {(L+M)^{2}}{l_{+}l_{-}}}(d\rho ^{2}+dz^{2})+e^{-2\psi (\rho ,z)}{\frac {L+M}{L-M}}\,\rho ^{2}d\phi ^{2}\,.}$

With the transformations[2]

${\displaystyle (9)\quad L+M=r\,,\quad l_{+}+l_{-}=2M\cos \theta \,,\quad z=(r-M)\cos \theta \,,}$

${\displaystyle \;\;\quad \rho ={\sqrt {r^{2}-2Mr}}\,\sin \theta \,,\quad l_{+}l_{-}=(r-M)^{2}-M^{2}\cos ^{2}\theta \,,}$

one can obtain the superposed Schwarzschild metric in the usual ${\displaystyle \{t,r,\theta ,\phi \}}$ coordinates,

${\displaystyle (10)\quad ds^{2}=-e^{2\psi (r,\theta )}\,{\Big (}1-{\frac {2M}{r}}{\Big )}\,dt^{2}+e^{2\gamma (r,\theta )-2\psi (r,\theta )}{\Big \{}\,{\Big (}1-{\frac {2M}{r}}{\Big )}^{-1}dr^{2}+r^{2}d\theta ^{2}\,{\Big \}}+e^{-2\psi (r,\theta )}r^{2}\sin ^{2}\theta \,d\phi ^{2}\,.}$

The superposed metric Eq(10) can be regarded as the standard Schwarzschild metric distorted by external Weyl sources. In the absence of distortion potential ${\displaystyle \{\psi (\rho ,z)=0,\gamma (\rho ,z)=0\}}$, Eq(10) reduces to the standard Schwarzschild metric

${\displaystyle (11)\quad ds^{2}=-{\Big (}1-{\frac {2M}{r}}{\Big )}\,dt^{2}+{\Big (}1-{\frac {2M}{r}}{\Big )}^{-1}\,dr^{2}+r^{2}\,d\theta ^{2}+r^{2}\sin ^{2}\theta \,d\phi ^{2}\,.}$

Similar to the exact vacuum solutions to Weyl's metric in spherical coordinates, we also have series solutions to Eq(10). The distortion potential ${\displaystyle \psi (r,\theta )}$ in Eq(10) is given by the multipole expansion[3]

${\displaystyle (12)\quad \psi (r,\theta )\,=-\sum _{i=1}^{\infty }a_{i}{\Big (}{\frac {R_{n}(\cos \theta )}{M}}{\Big )}P_{i}}$ with ${\displaystyle R:={\Big [}{\Big (}1-{\frac {2M}{r}}{\Big )}r^{2}+M^{2}\cos ^{2}\theta {\Big ]}^{1/2}}$

where

${\displaystyle (13)\quad P_{i}:=p_{i}{\Big (}{\frac {(r-m)\cos \theta }{R}}{\Big )}}$

denotes the Legendre polynomials and ${\displaystyle a_{i}}$ are multipole coefficients. The other potential ${\displaystyle \gamma (r,\theta )}$ is

${\displaystyle (14)\quad \gamma (r,\theta )\,=\sum _{i=1}^{\infty }\sum _{j=0}^{\infty }a_{i}a_{j}}$ ${\displaystyle {\Big (}{\frac {ij}{i+j}}{\Big )}}$ ${\displaystyle {\Big (}{\frac {R}{M}}{\Big )}^{i+j}}$${\displaystyle (P_{i}P_{j}-P_{i-1}P_{j-1})}$${\displaystyle -{\frac {1}{M}}\sum _{i=1}^{\infty }\alpha _{i}\sum _{j=0}^{i-1}}$ ${\displaystyle {\Big [}(-1)^{i+j}(r-M(1-\cos \theta ))+r-M(1+\cos \theta ){\Big ]}}$${\displaystyle {\Big (}{\frac {R}{M}}{\Big )}^{j}P_{j}\,.}$

• Weyl metrics
• Schwarzschild metric